Abstract
We generalize the Abel–Ruffini theorem to arbitrary dimension, i.e. classify general square systems of polynomial equations solvable by radicals. In most cases, they reduce to systems whose tuples of Newton polytopes have mixed volume not exceeding 4. The proof is based on topological Galois theory, which ensures non-solvability by any formula involving quadratures and single-valued functions, and the computation of the monodromy group of a general system of equations, which may be of independent interest.
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G. Gusev was partially supported by RFBR grant 13-01-00755.
This study (Research Grant No 14-01-0152) is supported by The National Research University—Higher School of Economics’ Academic Fund Program in 2014/2015. A. Esterov was partially supported by RFBR Grant 13-01-00755 and the Dynasty Foundation fellowship.
